Algebra

Let's first solve for ab:

$(a + b)^2 = a^2 + b^2 + 2ab$

$1^2 = 2 + 2ab$

$-1 = 2ab$

$ab = -\frac{1}{2}$

Now, we can solve for a^3 + b^3:

$(a^2 + b^2)(a + b) = a^3 + b^3 + a^2b + b^2a$

$(2)(1) = a^3 + b^3 + ab(a) + ab(b)$

$2 = a^3 + b^3 + ab(a + b)$

$2 = a^3 + b^3 + -\frac{1}{2}(1)$

$a^3 + b^3 = 2 + \frac{1}{2}$

$\mathbf{a^3 + b^3 = 5/2}$